Tubular neighborhood theorem

This fact is an application of the following pivotal fact/result/idea: inverse function theorem
View other applications of inverse function theorem OR Read a survey article on applying inverse function theorem

This fact is an application of the following pivotal fact/result/idea: existence of smooth partitions of unity
View other applications of existence of smooth partitions of unity OR Read a survey article on applying existence of smooth partitions of unity

This fact is an application of the following pivotal fact/result/idea: Lebesgue number lemma
View other applications of Lebesgue number lemma OR Read a survey article on applying Lebesgue number lemma

Statement

Let $M$ be a submanifold (differential sense) of $\mathbb {R} ^{n}$, of dimension $m$. Then, there exists $\epsilon >0$ such that for any point at distance at most $\epsilon$ from $M$, there is a unique expression of the point as a sum $p+v$ where $p\in M$ and $v$ is a normal at $p$, with $\|v\|<\epsilon$.

If we define $U$ as the open subset of $\mathbb {R} ^{n}$ comprising those points of $\mathbb {R} ^{n}$ at distance less than $\epsilon$ from $M$, then $U$ can be viewed as a concrete realization, in the ambient space $\mathbb {R} ^{n}$, of the normal bundle to $M$ in $\mathbb {R} ^{n}$. In the situations where the normal bundle to $M$ is trivial, we see that this gives a natural diffeomorphism $U\cong M\times \mathbb {R} ^{n-m}$.